Poloidal and toroidal plasma rotation and resistive wall modes in tokamaks

摘要

Contrary to wide spread belief, determination of the frequencies and growth rates of MHD waves and instabilities in rotating plasmas does not require non-self adjoint operators. The physics involves the generalized force operator and the Doppler-Coriolis shift operator, which are both self-adjoint, but they occur in a quadratic eigenvalue problem [1] with complex eigenvalues. Enclosing the system with a resistive wall yields a cubic eigenvalue problem [2], where the dissipation of the wall permits the additional class of resistive wall modes. Since these modes grow on a much longer time scale than the ideal MHD ones, they may be feedback stabilized [3], To accomplish that', knowledge of the full spectrum of modes of the system is essential. A general method to achieve this for the quadratic (ideal) eigenvalue problem has been developed recently by constructing the solution paths in the complex ω-plane [4]. These are obtained by taking away the outer boundary and solving the open boundary value problem, but restricting the solutions to have no energy flow into or out of the system. This yields curves in the complex plane on which the eigenvalues must be situated. They are determined by imposing the missing boundary condition. Here, the method is generalized to the cubic (dissipative) eigenvalue problem by accounting for the energy dissipation in the resistive wall. The obtained topologies of the solution paths yield important new insights into the. coupling of the resistive wall modes with the co- and counter-rotating external kink modes. Stability regimes obtained depend on the details of the profiles of the safety factor q, the toroidal velocity vψ, the poloidal velocity v_p, the wall position and the dissipative time scale tD of the wall.